Suppose $R_{1}$ and $R_{2}$ are relations on $A$. If $R_{1}$ and $R_{2}$ are transitive, must $R_{1} \backslash R_{2}$ be transitive?

Question

The following is counterexample I read but, I don't understand how $R_{2}$ is vacuously transitive. Can someone please explain.

COUNTEREXAMPLE:

Let $A= \left\{1,2,3 \right\}$ and define $R_1 =\left\{(1,2),(2,3),(1,3) \right\}$ this is clearly transitive and $R_2=\left\{(1,3)\right\}$ is vacuously transitive.

But $R_1 \backslash R_2 = \left\{(1,2),(2,3)\right\}$ which is not transitive.

Answer 1

Transitivity is a "contract" that says something about what we require in a relation whenever we find two pairs in the relation such that the right element of one pair is equal to the left element of the other. We can't find two such pairs in $R_2$, which means it can't possibly break the contract, and it is therefore transitive. "Vacuously" is a word we use to emphasise that it is transitive precisely because we can never find two such pairs, and thus the contract can never even really be tested.